Problem: Given $ \overrightarrow{BA}\perp\overrightarrow{BD}$, $ m \angle ABC = 7x - 52$, and $ m \angle CBD = 4x + 21$, find $m\angle CBD$. $B$ $A$ $D$ $C$
Explanation: From the diagram, we see that together ${\angle ABC}$ and ${\angle CBD}$ form ${\angle ABD}$ , so $ {m\angle ABC} + {m\angle CBD} = {m\angle ABD}$ Since we are given that $\overrightarrow{BA}\perp\overrightarrow{BD}$ , we know ${m\angle ABD = 90}$ Substitute in the expressions that were given for each measure: $ {7x - 52} + {4x + 21} = {90}$ Combine like terms: $ 11x - 31 = 90$ Add $31$ to both sides: $ 11x = 121$ Divide both sides by $11$ to find $x$ $ x = 11$ Substitute $11$ for $x$ in the expression that was given for $m\angle CBD$ $ m\angle CBD = 4({11}) + 21$ Simplify: $ {m\angle CBD = 44 + 21}$ So ${m\angle CBD = 65}$.